We define a real number $x$ to be pseudopositive if $\forall y \in \mathbb{R}$ we have $ \neg \neg (x > y) \vee \neg \neg (y > 0) $.
The Weak Markov's Principle (WMP) is the axiom that every pseudopositive real number is a positive real number. This clearly follows from the analytic Markovs Principle which states that the double negations above simplify, so it is clearly true in the effective topos and in any CC + MP setting. The idea behind the generalization is that the WMP apparently holds in intuitionistic models with a continuity principle.
This then makes me wonder for which sheaf models it holds. Is there a sheaf model where it fails? Does it hold for say, sheaves over manifolds?
EDIT: Translating this question to a standard one for real functions on a topological space which is a standard question about topology, the question is for what topological spaces $X$ we have the following property:
We can define a continuous real function $f(x)$ to be pseudopositive if for any continuous real function $g(x)$ on $X$ we have
$\textrm{int} (\textrm{cl}(\{ x: f(x) > g(x) \})) \bigcup \textrm{int} (\textrm{cl}(\{ x: g(x) > 0 \})) = X $
and we may say that the space $X$ satisfies the weak markov property if the pseudopositive real functions on $X$ are just the positive functions on $X$.
A quick sanity check tells is that this is automatically true if every open subset is clopen, which includes discrete spaces and stone spaces like the Cantor space/p-adic numbers or the Irrational numbers, or more generally if they satisfy that the interior of the closure of the open support of a continuous function simplifies (analytic MP), but we are interested in whether spaces that look more like $\mathbb{R}^n$ satisfy this weaker property.
This question is fairly complicated, but here's a tl;dr:
By the usual semantics for $\mathsf{Sh}(\mathbb{R}^n)$, thm 1 holds if and only if, for each open $U \subseteq \mathbb{R}^n$, whenever $U \Vdash \ulcorner f \text{ is pseudopositive} \urcorner$ then $U \Vdash f > 0$ (See Mac Lane & Moerdijk, Sheaves in Geometry and Logic pg 316 for this theorem). Notice this implication is at the level of the metatheory, so we're allowed to use proof by contradiction, etc. without worry.
With this in mind, fix an open $U \subseteq \mathbb{R}^n$, and assume that $U \not \Vdash f > 0$. We'll show that $U \not \Vdash \ulcorner f \text{ is pseudopositive} \urcorner$.
Now $U \not \Vdash f > 0$ means that $f$ is not strictly positive on $U$. If there's a point with $f(p) < 0$ then $f$ has a neighborhood on which $f < 0$, so clearly is not pseudopositive. So we may assume $f \geq 0$ on $U$, but is not strictly positive. Fix a $p \in U$ with $f(p) = 0$. We'll build a function $g$ so that $U \not \Vdash \lnot \lnot (g > 0) \lor \lnot \lnot (f > g)$. To do this, we need to know $g \leq 0$ on an open set and $f \leq g$ on an open set. Moreover, since truth in a sheaf topos is local, we'll need to know that we can't get our way out of this problem by moving closer to $p$. So we'll need these properties to hold on every neighborhood of $p$.
Some thought shows that $g(x) = 2f(x) \sin \left ( \frac{1}{\lVert x-p \rVert} \right )$ does the trick. Since $f \geq 0$, we see $g \leq 0$ whenever $\sin \left ( \frac{1}{\lVert x-p \rVert} \right ) \leq 0$, which happens on an open set in every neighborhood of $p$. Moreover, whenever $\sin \left ( \frac{1}{\lVert x-p \rVert} \right ) \geq \frac{1}{2}$ (which also happens on an open set in every neighborhood of $p$) we'll have $g \geq f$! The only question is whether $g$ is continuous, but since $f(p) = 0$ the squeeze theorem tells us that taking $g(p) = 0$ gives a continuous function on $U$.
So then if $f$ is any function where $U \not \Vdash f > 0$, we also have $U \not \Vdash \ulcorner f \text{ is pseudopositive} \urcorner$, so (using LEM in the metatheory) $U \Vdash \ulcorner f \text{ is pseudopositive} \urcorner$ implies $U \Vdash f > 0$ for every $U \subseteq \mathbb{R}^n$, as desired.
In pictures, given some blue function $f$ which is $0$ at the $p$, we build an orange function $g$ which, in any neighborhood of $p$, has both $g \geq f$ and $g \leq 0$ on an open set. This witnesses the non-pseudopositivity of $f$ under the assumption that $f(p) = 0$ somewhere.
Now, since truth is local in a sheaf topos, we immediately get a corollary:
For the proof, notice that we can open cover $X$ by spaces homeomorphic to $\mathbb{R}^n$. Since each element of the cover validates WMP by Thm $1$, we see that $X$ does as well.
I've actually had this answer drafted for over a week now, but I really wanted to find a space invalidating WMP. After some googling, I found Diener's Constructive Reverse Mathematics, which in section 7.2.5 gives the result I promised at the top of this answer.
Fix a nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$. Let $X = \mathbb{N} \cup \{ \omega \}$ where a base of the topology is given by
Then consider the real $f(n) = 2^{-n}$, $f(\omega) = 0$. You can check this is pseudopositive, but it is not positive since $f(\omega) = 0$.
Edit: Here's a proof that $f$ is pseudopositive. Let $g$ be some other function. Then exactly one of $\{ n \mid g(n) > 0 \}$ or $\{ n \mid g(n) \leq 0 \}$ is $\mathcal{U}$-big. Note that if $A$ is $\mathcal{U}$-big, then $A \cup \{ \omega \} \Vdash \lnot \lnot \varphi$ exactly when $\varphi$ holds on each $n \in A$, but possibly not on $\omega$.
If $A = \{ n \mid g(n) > 0 \}$ is $\mathcal{U}$-big, then look at the open cover $\Big \{ A \cup \{ \omega \}, \{ n \} \text{ for each $g(n) \leq 0$} \Big \}$. On each $\{ n \}$ we have $f > g$, and on $A \cup \{ \omega \}$ we have $\lnot \lnot (g > 0)$ (since $g > 0$ is true on $A$). So $\mathsf{Sh}(X) \vDash \lnot \lnot (g > 0) \lor \lnot \lnot (f > g)$. If instead $B = \{ n \mid g(n) \leq 0 \}$ is $\mathcal{U}$-big, then we get an open cover $\Big \{ B \cup \{ \omega \}, \{ n \} \text{ for each $g(n) > 0$} \Big \}$. Now on each $\{ n \}$ we have $g > 0$, and on $B \cup \{ \omega \}$ we have $\lnot \lnot (f > g)$ (since $f > 0 \geq g$ is true on $B$). So again we have $\mathsf{Sh}(X) \vDash \lnot \lnot (g > 0) \lor \lnot \lnot (f > g)$, as desired.
In fact, this paper has a ton of interesting results about the internal logic of sheaf topoi, so you'll almost certainly enjoy giving it a read! I know I did.
I hope this helps ^_^